On Relativistic Particle Creation in Bose-Einstein Condensates

On relativistic particle creation in Bose-Einstein condensates Carlos Sab´ın and Ivette Fuentes School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom Abstract We show that particle creation of Bogoliubov modes in a Bose-Einstein condensate due to the accelerated motion of the trap is a genuinely relativistic effect. To this end we show that Bogoliubov modes can be described by a time rescaling of the Minkowski metric. A consequence of this is that Rindler transformations are perceived by the phonons as gen- eralised Rindler transformations where the speed of light is replaced by the speed of sound, enhancing particle creation at small velocities. Since the non-relativistic limit of a Rindler transformation is just a Galilean transformation entailing no length contraction or time dilation, we show that the effect vanishes in the non-relativistic limit. Introduction Bose-Einstein condensates (BEC) are natural testbeds for analogue gravity scenarios [1]. By artificially changing the parameters of the condensate, several spacetime metrics such as black holes or expanding universes [2] can be simulated while the real spacetime metric remain flat. Recently, we and our collaborators have initiated a new avenue of research in which we have shown that changes of the real spacetime metric such as real acceleration or gravitational waves are able to generate observable effects in the BEC [3, 4, 5], which can be relevant in quantum information space-based experiments and can be used as working principle of ultra sensitive quantum measurement devices. Although related, both programmes of research are neatly different. Since a BEC is commonly treated as a non-relativistic system and the relevant scale of speed is provided by the speed of sound -which is typically as low as a few mm/s- arXiv:1405.5789v1 [quant-ph] 22 May 2014 it is tempting to think that there could not possibly be relativistic phenomena in these systems. It is the aim of this work however to clearly discriminate between analogue and genuine relativistic effects in BECs and indisputably confirm that the phenomena described in [3, 4, 5] belong to the latter. In order to build on solid grounds it is obviously necessary to first state what we mean by relativistic. It is a widespread belief that an effect is relativistic only if it occurs at velocities close to the speed of light. However, technological developments have enabled the measurement of a paradigmatic special relativity phenomenon such as time dilation at velocities as low as 10 m/s [6]. The accelerated rate at which state-of-the-art quantum metrology devices evolve, promises to achieve unforeseeable levels of accuracy and precision. 1 It thus looks more rigorous to define relativistic effect as something that must be described within a framework consistent with the postulates of Einstein’s relativity, regardless the scale of speeds involved in the experiment. That is the case in [6]: Lorentz transformations are required to reproduce the experimental results, which cannot be explained by means of Galilean transformations. In this work we will show that this is indeed the case for the generation of particles and mode-mixing described in [3, 4, 5]: they can only be properly described by Rindler transformations- a particular instance of Lorentz transformations- and vanish if Galilean transformations are considered. The effective metric of Bogoliubov modes We start by describing the BEC on a general spacetime metric following references [7, 8]. In the superfluid regime, a BEC is described by a mean field classical background Ψ plus quantum fluctuations Π.ˆ These fluctuations, for length scales larger than the so-called healing length, behave like a phononic quantum field on a curved metric. Indeed, in a homogenous condensate, the field obeys a massless Klein-Gordon equation Πˆ = 0 (1) where the d’ Alembertian operator ab =1/√ g ∂a(√ gg ∂b) (2) − − depends on an effective spacetime metric gab -with determinant g- given by [7, 8] 2 −1 2 n0 cs cs gab = gab + 1 2 VaVb . (3) ρ0 + p0 − c The effective metric is a function of the real spacetime metric gab (that in general may be curved) and background mean field properties of the BEC such as the number density n0, the energy density ρ0, the pressure p0 and the speed of sound ∂p cs = c . (4) s∂ρ Here p is the total pressure, ρ the total energy density and Va is the 4-velocity flow on the BEC. This description stems from the theory of fluids in a general relativistic background [7], and thus is valid as long as the BEC can be described as a fluid, that is as long as it remains within the quantum hydrodynamic regime [8]. In the field of analogue gravity, the real spacetime metric is considered to be flat and, therefore, its effects are neglected. Analogue spacetimes are simulated through the artificial manipulation of what we call the analogue gravity metric, 2 −1 2 n0 cs cs Gab = 1 2 VaVb . (5) ρ0 + p0 − c Experimentalists change the background parameters of the analogue metric Gab to simulate sonic black holes or expanding universes [1, 2]. Here we are interested solely on the effects of the real spacetime metric. To ensure this, we consider that in the comoving frame V = (c, 0, 0, 0) (6) 2 and obtain, 2 2 (c cs) 00 0 2 −1 − n0 cs 0 000 gab = gab + . (7) ρ0 + p0 0 000 0 000 If the real spacetime metric is flat, then gab = ηab, where ηab c2 0 0 0 −0 100 η = . (8) ab 0 010 0 001 c is the speed of light in the vacuum and we consider Minkowski coordinates (t,x,y,z). Therefore, the effective metric of the BEC phononic excitations on the flat spacetime metric is given by, 2 cs 0 0 0 2 −1 − n0 cs 0 100 gab = . (9) ρ0 + p0 0 010 0 001 Ignoring the conformal factor -which can always be done in 1D or in the case in which is time-independent- we notice that the metric is the flat Minkowski metric with the speed of light being replaced by the speed of sound cs. Notice that considering a rescaled time coordinate: c t′ = t ; x′ = x ; y′ = y ; z′ = z (10) cs we recover the standard Minkowski metric in Eq. (8) from the one in Eq. (9). This means that the phonons live on a spacetime that is Minkowski however, due to the BEC ground state properties, time flows at a different rate and excitations propagate accordingly. As a result of this, changes in the real spacetime metric are amplified, becoming observable. Rindler transformations The aim of this section is the analysis of Rindler transformations of coordinates and, in particular, their effects on the Bogoliubov modes. Rindler coordinates are suitable for uniformly accelerated observers. For the sake of simplicity and without loss of generality, we will consider a 1+1 dimensional spacetime in what follows. Thus, the line element of Minkowski spacetime is: 2 2 2 2 ds = c dt + dx . (11) − Now, we consider a Rindler transformation: χ t = sinh η; x = χ cosh η. (12) c An uniformly accelerated observer with proper acceleration a is static in Rindler coordinates, that is she follows a trajectory of constant Rindler position χ0: c2 χ0 = . (13) a 3 The proper time τ of such observer is χ0 η τ = . (14) c The line element transforms under the Rindler transformation into: 2 2 2 2 ds = c dτ + dχ . (15) − By using Eqs. (10) and (12) we can relate the phonon coordinates (t′, x′) with the Rindler ones (η,χ). It is easy to find that: χ t′ = sinh η; x′ = χ cosh η. (16) cs Therefore, Rindler transformations are seen by the phonons as Rindler transformations that depend on cs instead of c. The line element of the phonons is conformal to: ds2 = c2 dt′2 + dx′2, (17) − s and it is transformed under a Rindler transformation to: ds2 = c2 dτ ′2 + dχ2, (18) − s where τ ′ is the proper time given by the line element in Eq. (17) of an observer following a trajectory χ = χ0: ds χ0 η τ ′ = dτ = = , (19) c √ g00 cs Z Z − 2 −cs where g00 = c2 is the corresponding element of the metric defined by the line element in Eq. (17): 2 µ ν µ ′ ′ ds = gµν dx dx , x = (ct , x ). (20) The proper acceleration a of an observer with proper time τ ′ is: 2 2 µ µ ν cs µ d x a = a gµν a = ; a = 2 . (21) | | χ0 d τ ′ p Putting all the above together, we see that the description of Bogoliubov modes, both for inertial and uniformly accelerated observers, is exactly the same as photons but replacing everywhere the speed of light c by the speed of sound cs. In particular, the change in the state of phonons in a cavity due to a sudden change in acceleration is quantified by the Bogoliubov coefficients ∗ αmn = (φˆm, φn) , βmn = (φˆn, φ ) (22) − m which relate the set of inertial modes φ with the uniformly accelerated ones φˆ through the Klein-Gordon inner product [9]. The inertial field operators ak are transformed into, ∗ ∗ † aˆm = αmnan + βmnan , (23) n X † wherea ˆk anda ˆk are creation and annihilation operators associated to the accelerated mode solutions. Then the coefficients αmn characterise the mode-mixing 4 induced by the transition from inertial to accelerated motion while βmn rep- resent particle creation. In the case of photons in a cavity, αmn, βmn can be computed analytically [10] as a series expansion in the parameter h, which is: aL h = , (24) c2 where a is the proper acceleration of an observer in the centre of the cavity and L is the constant proper length of the cavity in the Rindler frame, L = χR χL (25) − The larger the value of h, the larger the mode-mixing and particle creation created by the inertial-to-accelerated transition.

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